What is the general solution for 3cos^2(θ)-sin(θ)=1 ?

The general solution for 3cos^2(θ)-sin(θ)=1 is θ=(3pi)/2+2pin,θ=0.72972…+2pin,θ=pi-0.72972…+2pin

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3cos^2(θ)-sin(θ)=1

3 cos^{2} (θ) - sin (θ) = 1

θ = \frac{3 π}{2} + 2 πn, θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

Solution steps

3 cos^{2} (θ) - sin (θ) = 1

Subtract

1

from both sides

3 cos^{2} (θ) - sin (θ) - 1 = 0

Rewrite using trig identities

2 - sin (θ) - 3 sin^{2} (θ) = 0

Solve by substitution

sin (θ) = - 1, sin (θ) = \frac{2}{3}

sin (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

sin (θ) = \frac{2}{3} : θ = arcsin (\frac{2}{3}) + 2 πn, θ = π - arcsin (\frac{2}{3}) + 2 πn

Combine all the solutions

θ = \frac{3 π}{2} + 2 πn, θ = arcsin (\frac{2}{3}) + 2 πn, θ = π - arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

θ = \frac{3 π}{2} + 2 πn, θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

5 cos (2 x) = 9 sin (x) + 4

- 3 cos (x) - 1 = cos (2 x)

- 2 tan^{2} (θ) - 4 tan (θ) + 2 = 3 tan (θ) + 5

csc (3 x) = 1

x, cos (x + y) + cos (x) = 0

What is the general solution for 3cos^2(θ)-sin(θ)=1 ?
The general solution for 3cos^2(θ)-sin(θ)=1 is θ=(3pi)/2+2pin,θ=0.72972…+2pin,θ=pi-0.72972…+2pin